Scatter Plots

Learn how to make a scatter plot from a table and describe the pattern: form, direction, strength, clusters, gaps and outliers.

Intermediate25 min18 problems
Practice Problems

Definition

A scatter plot is a graph that shows the relationship between two numerical variables measured on the same individuals. Each individual becomes one point (x,y)(x, y).
  • The explanatory variable (the one that may explain or influence the other) goes on the **horizontal xx-axis**.
  • The response variable (the one that responds) goes on the **vertical yy-axis**.
To make a scatter plot from a table:
  1. 1.Decide which variable is xx and which is yy.
  2. 2.Choose a scale for each axis that covers all the values with equal steps.
  3. 3.Label both axes with the variable name and unit.
  4. 4.Plot one point for every row of the table. Do not connect the points.
To describe a scatter plot, look at:
  • Form: do the points follow a straight line (linear) or a curve (non-linear)?
  • Direction: going up from left to right (positive) or down (negative)?
  • Strength: are the points close to the pattern (strong), loosely around it (moderate) or widely spread (weak)?
  • Unusual features: clusters (groups of points), gaps (empty regions) and outliers (points far from the pattern).

Worked Examples

A basketball coach records how many hours seven players practice per week and how many free throws (out of 20) each one makes:
PlayerABCDEFG
Hours1234567
Free throws68911121415
 Which variable goes on each axis, and what scales should you use?
1

Find the explanatory variable

Practice may influence the number of free throws, not the other way round → xx = hours of practice, yy = free throws made
2

Find the range of each variable

Hours: from 11 to 77. Free throws: from 66 to 1515 → xx needs to reach at least 77, yy at least 1515
3

Choose equal steps

xx-axis: 00 to 88 in steps of 11. yy-axis: 00 to 2020 in steps of 22 (nobody can make more than 20 free throws, and 20÷2=1020 \div 2 = 10 equal steps) → Both axes have equal spacing
4

Label the axes

xx: Practice per week (hours). yy: Free throws made (out of 20) → Axes are ready for plotting

Common Mistakes

Putting the response variable on the xx-axis

Why it's wrong: The axes have different jobs. If you plot free throws on the xx-axis and practice on the yy-axis, the graph suggests that free throws explain practice time.

Correct: Ask: which variable could influence the other? That one is the explanatory variable and goes on the xx-axis.

Using uneven steps on an axis (for example 0, 5, 10, 20, 50)

Why it's wrong: Uneven spacing squeezes some points together and stretches others apart, so the pattern you see is false.

Correct: Use the same step all along each axis, for example 0, 10, 20, 30, 40, 50. The two axes may have different steps.

Connecting the points with lines like a line graph

Why it's wrong: The rows of the table are separate individuals, not moments in time. Joining them invents a path that does not exist.

Correct: Leave the points as separate dots. Describe the overall pattern in words (in the next lesson you will draw one line of best fit).

Thinking a steep pattern is a strong pattern

Why it's wrong: Steepness tells you how fast yy changes. Strength tells you how closely the points follow the pattern. A pattern with a gentle slope can be very strong, a steep one can be weak.

Correct: Judge strength by how tightly the points hug the line or curve, not by how steep it is.

Interactive Visual

Scatter Plot

Practice hours vs free throws made

0246805101520Practice per week (hours)Free throws made (out of 20)ABCDEFG

Tap a point to see its coordinates.

Scatter Plot

Homework time vs quiz score: clusters, a gap and an outlier

0204060800246810Homework time (minutes)Quiz score (out of 10)Outlier

Tap a point to see its coordinates.

Function graph

Plot these points:

  • (1, 2)
  • (2, 3)
  • (3, 5)
  • (4, 6)
  • (5, 7)
  • (6, 9)
12345678123456789100People in the householdLaundry loads per week

Tap the grid to place a point. Tap a point again to remove it.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems
Problem 1 of 18
Easy
A gardener studies how the hours of sunshine per day affect the growth of tomato plants. Which variable goes on the xx-axis of the scatter plot?

Why It Matters

Scatter plots are the first thing a scientist, coach or analyst draws when two measurements might be connected:
  • Sports: A coach plots practice hours against points scored to see if training pays off
  • Health: Doctors plot age against blood pressure to see how it changes over a lifetime
  • Weather: Meteorologists plot altitude against temperature (it drops about 6°C6°C for every kilometer up)
  • Business: A shop plots temperature against ice cream sales to plan how much to order
  • Science: Physicists plot speed against braking distance and discover it is a curve, not a straight line
A scatter plot shows the whole picture at once: the general pattern, the points that break it and the groups hiding in the data. It is the starting point for correlation and for the line of best fit in the next lessons.

Real World Applications

Sports Science: Height and Arm Span

Coaches in swimming and basketball plot height against arm span, because a long arm span is an advantage in both sports.

Example:

For most people the arm span is almost equal to the height, so the points lie close to a straight line going up: a strong, positive, linear pattern.

1Try It Yourself

You want to plot the arm spans of a team. They range from 148148 cm to 182182 cm. Your yy-axis starts at 140140 cm and has 1010 grid squares.

What step should each grid square represent (choose from the nice steps 1, 2, 5 or 10)?

Step 1: Write the mathematical expression

Enter the step for one grid square:

Weather: Altitude and Temperature

Mountain guides and pilots know that the air gets colder the higher you go. A scatter plot of altitude against temperature shows how fast.

Example:

Weather balloons measure the temperature every few hundred meters on the way up. Plotted together, the readings fall almost on a straight line going down.

2Try It Yourself

A hiker records the temperature at four altitudes: (0,20)(0, 20), (1,14)(1, 14), (2,7)(2, 7), (3,1)(3, 1), with altitude in km and temperature in °C.

How would you describe the scatter plot?

Step 1: Choose the right answer

Choose the best description:

Business: Finding an Outlier

Shops plot the number of social media posts per week against the number of new customers to see if advertising works, and to spot weeks that do not fit.

Example:

A week with many posts but very few customers stands out as an outlier. It often has a reason: a holiday, a website problem or bad weather.

3Try It Yourself

A bakery records five weeks as (posts, new customers): (2,30)(2, 30), (4,42)(4, 42), (6,54)(6, 54), (8,66)(8, 66), (10,20)(10, 20).

What is the number of posts in the week that is an outlier?

Step 1: Write the mathematical expression

Enter the xx-value of the outlier:

Key Takeaways

  • 1A scatter plot shows two numerical variables: each individual is one point (x,y)(x, y)
  • 2The explanatory variable goes on the xx-axis, the response variable on the yy-axis
  • 3Each axis needs equal steps, a label and a unit; the two axes may use different scales
  • 4Plot one dot per row and never connect the dots
  • 5Describe the pattern by form (linear or non-linear), direction (positive or negative) and strength (strong, moderate, weak)
  • 6Look for unusual features: clusters, gaps and outliers

Frequently Asked Questions

A line graph follows one quantity over time, so joining the points makes sense. A scatter plot compares two variables for many separate individuals, so the points stay unconnected.
A line graph follows one quantity over time, so joining the points makes sense. A scatter plot compares two variables for many separate individuals, so the points stay unconnected.
No. If all heights are between 148 and 182 cm, starting the axis at 140 uses the space better. Just keep equal steps and make the starting value clear on the axis.
Sometimes, like with height and arm span, neither one causes the other. Then you may choose either variable for the xx-axis. Pick the one you would most likely use to estimate the other.
Not automatically. First check whether it is a mistake in the data. If it is a real value, keep it and mention it in your description: outliers often tell the most interesting story.

Glossary

Scatter plot
A graph of points (x,y)(x, y) that shows the relationship between two numerical variables
Explanatory variable
The variable that may explain or influence the other one; it goes on the xx-axis
Response variable
The variable that responds to the explanatory variable; it goes on the yy-axis
Form
The shape of the pattern: linear (straight line) or non-linear (curve)
Cluster
A group of points that lie close together, separated from the rest
Outlier
A point that lies far from the overall pattern of the other points

Formula Card

One point per individual

(x,y)=(explanatory,response)(x, y) = (\text{explanatory}, \text{response})

Every row of the table becomes one point: the first value goes along, the second goes up

Range of a variable

range=largest−smallest\text{range} = \text{largest} - \text{smallest}

Tells you how much of the axis the values need

Size of one step

step≥largest−startnumber of squares\text{step} \geq \frac{\text{largest} - \text{start}}{\text{number of squares}}

Round up to a nice step such as 1, 2, 5 or 10

Today's challengeOne problem, three tries

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